Description: Show that the original axiom ax-c4 can be derived from ax-4 ( alim ), ax-10 ( hbn1 ), sp and propositional calculus. See ax4fromc4 for the rederivation of ax-4 from ax-c4 .
Part of the proof is based on the proof of Lemma 22 of Monk2 p. 114. (Contributed by NM, 21-May-2008) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc4 | ⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sp | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 𝜑 ) | |
| 2 | 1 | con2i | ⊢ ( ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
| 3 | hbn1 | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) | |
| 4 | hbn1 | ⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) | |
| 5 | 4 | con1i | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) |
| 6 | 5 | alimi | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) |
| 7 | 2 3 6 | 3syl | ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) |
| 8 | alim | ⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) | |
| 9 | 7 8 | syl5 | ⊢ ( ∀ 𝑥 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) |